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Unformatted text preview: I E/Stat 265, Fall 2009 I E/Stat 265, Fall 2009 O /Stat 65, a 009 O /Stat 65, a 009 Lecture #15: Lecture #15: Estimation Concepts and Methods Estimation Concepts and Methods- Concepts of Point Estimation s a o C o p s a d o d s s a o C o p s a d o d s 6 1 Concepts of Point Estimation A. Biased Vs. Unbiased Estimators B. Minimum Variance Estimators C. Standard Error of a Point Estimate 6-2 Point Estimation Methods A. Method of Moments (MOM) 1 B. Method of Maximum Likelihood (MLE) C. Method of Bayes (Not testable this term) 1 Point Estimates 6.1 Point Estimates Objective obtain an estimate of a population parameter from a sample (e.g. sample mean, , is point estimate of population mean x ) x Applications: Parameter Estimation Hypothesis Testing (and other inferential statistics methods) 2 arameter Estimation Example Parameter Estimation Example uppose you wish to estimate the population Suppose you wish to estimate the population mean, ,. Some possible estimators include: Sample Mean, Sample Median, Sample Trimmed Mean, Sample Geometric Mean, Sample Harmonic Mean, (Max X + Min X)/2 Recall example from descriptive statistics, which of the following is the best estimator? ample Mean = 965 0 Sample Median = 1009 5 Sample Mean = 965.0 Sample Median = 1009.5 Sample Trim Mean = 971.4 3 arameter Estimates it Parameter Estimates - Fit Estimates, , always have measurement error. hy? estimation error = + Why? Because is a function of observed x i s (r.v.) that change with n We identify the best estimator as the one with: inimum Bias (unbiased) vg stimation error Minimum Bias (unbiased) avg estimation error = 0 Minimum Variance of estimation error 0 as n More consistent predictions 4 p Increases the likelihood the parameter estimate represents the true parameter iased Vs Unbiased Estimator Biased Vs. Unbiased Estimator Bias - difference between the expected value of the statistic and the true parameter nbiased Estimator: Bias = E( ) Unbiased Estimator: Example: is an unbiased estimated of (Bias = 0) Suppose X 1 , X 2 , .. X n are iid rvs, with E(x i ) = x n i i 1 E(x ) n E(x) E( ) = = = = = 9 n n nbiased Estimator of Variance Unbiased Estimator of Variance hich of the following formulas is an unbiased Which of the following formulas is an unbiased estimator of variance, 2 ? n n 2 2 i i 2 2 i 1 i 1 1 2 (X X) (X X) s n 1 n = = = = = = ( ) ( ) = = 2 2 2 2 n 1 E s E n What happens to the bias effect as n becomes large?...
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This note was uploaded on 03/17/2010 for the course IOE 265 taught by Professor Garyherrin during the Fall '09 term at University of Michigan-Dearborn.
- Fall '09